A Shape Dynamical Approach to Holographic Renormalization

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provided by Open Access Repository Eur. Phys. J. C (2015) 75:3 DOI 10.1140/epjc/s10052-014-3238-z

A shape dynamical approach to holographic renormalization

Henrique Gomes1,a, Sean Gryb2,3,b, Tim Koslowski4,c, Flavio Mercati5,d, Lee Smolin5,e 1 University of California at Davis, One Shields Avenue, Davis, CA 95616, USA 2 Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 3 Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Nijmegen, The Netherlands 4 University of New Brunswick, Fredericton, NB E3B 5A3, Canada 5 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada

Received: 7 October 2014 / Accepted: 18 December 2014 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We provide a bottom-up argument to derive some 1.3 Main results ...... known results from holographic renormalization using the Notation ...... classical bulk–bulk equivalence of General Relativity and 2 Mechanism ...... Shape Dynamics, a theory with spatial conformal (Weyl) 2.1 The basic idea of gauge symmetry trading in invariance. The purpose of this paper is twofold: (1) to adver- General Relativity ...... tise the simple classical mechanism, trading off gauge sym- 2.2 Shape Dynamics description of gravity ...... metries, that underlies the bulk–bulk equivalence of General 2.3 Special CMC slices ...... Relativity and Shape Dynamics to readers interested in dual- 3 Asymptotically locally AdS and VPCT ...... ities of the type of AdS/conformal field theory (CFT); and 4 Holographic renormalization and Shape Dynamics .. (2) to highlight that this mechanism can be used to explain 4.1 Classical Shape Dynamics at large volume .... certain results of holographic renormalization, providing an 4.1.1 Volume expansion ...... alternative to the AdS/CFT conjecture for these cases. To 4.1.2 Hamilton–Jacobi equation ...... make contact with the usual semiclassical AdS/CFT corre- 4.2 Comparison with holographic renormalization spondence, we provide, in addition, a heuristic argument that results ...... makes it plausible that the classical equivalence between 5 Remarks on Wilsonian renormalization ...... General Relativity and Shape Dynamics turns into a dual- 5.1 Generic recipe ...... ity between radial evolution in gravity and the renormaliza- 6 Conclusions ...... tion group flow of a CFT. We believe that Shape Dynamics Appendix A: general mechanism for gauge symmetry provides a new perspective on gravity by giving conformal trading ...... structure a primary role within the theory. It is hoped that this A.1. Symmetry trading and linking gauge theories .. work provides the first steps toward understanding what this A.2. Dictionary and observable equivalence ...... new perspective may be able to teach us about holographic A.3. Construction of linking gauge theories ...... dualities. Appendix B: scalar field (d ≤ 4) ...... References ......

Contents 1 Introduction 1 Introduction ...... 1.1 Heuristics ...... A key insight of holography is that the behavior of a con- 1.2 Shape Dynamics ...... formal ﬁeld theory (CFT) under renormalization is gov- erned by a diffeomorphism-invariant gravitational theory in a e-mail: [email protected] one higher spacetime dimension. In this correspondence, the b e-mail: [email protected] extra dimension plays the role of the renormalization scale. c e-mail: [email protected] This gives the renormalization group a geometric setting and d e-mail: [email protected] suggests deep connections between the dynamics of space- e e-mail: [email protected] time and the renormalization of quantum ﬁelds. 123 3 Page 2 of 15 Eur. Phys. J. C (2015) 75:3

The holographic formulation of the renormalization group which allows us to posit correspondences between them? was elucidated in a series of beautiful papers [1–9] and under- One well-known answer builds on global symmetries, in par- lies much of the very fertile applications of holography and ticular, the correspondence between the AdS group in d + 1 the AdS/CFT correspondence [10–12] to condensed matter dimensions and the global conformal group in d dimensions. and fluid systems. In this paper, we offer new insights into Here we would like to suggest that a deeper correspondence the connection between gravitation and renormalization by holds between gauge symmetries, in particular between the giving a new explanation for why that connection exists. We group of refoliations of the spacetime manifold in d + 1 give a novel derivation of a correspondence between a CFT dimensions and the group of Weyl transformations acting to defined on a d-dimensional manifold with metric Gab, locally rescale the metric on its boundary. This connection and a solution to a gravitational theory in d + 1 dimensions. between refoliations and Weyl transformations is captured In this correspondence, is to be interpreted as the asymp- by the mechanism of gauge symmetry trading which is at totic boundary of a Euclidean asymptotically AdS spacetime, the heart of Shape Dynamics. The main idea we want to M, the latter being a solution to a spacetime diffeomorphism develop in this paper is that it is the existence of gauge sym- invariant theory of a metric gμν such that = ∂M and Gab metry trading between refoliations of a d + 1 dimensional is the pullback of gμν onto .1 We give a plausibility argu- spacetime and Weyl transformations on fixed foliations of ment for the conjecture that this correspondence can be seen that spacetime that provide a deep and very general reason as the reason why gravitational evolution encodes the renor- why there exist correspondences between bulk gravitational malization group flow of a CFT, at least to leading order theories and boundary CFTs.2 in a semiclassical regime, near the conformal boundary of a The main work of this paper consists of showing how Euclidean locally asymptotically AdS spacetime. trading of spacetime refoliation invariance for spatial Weyl The new derivation we offer makes use of a recent refor- invariance can be used to explain a subset (which will mulation of General Relativity in which spatial conformal be explicitly specified later) of the results of holographic invariance plays a central role. This formulation, called renormalization. This trading of spacetime refoliation invari- Shape Dynamics, is the result of trading the many fingered ance for spatial Weyl invariance has some immediate con- time aspect of spacetime diffeomorphism invariance—which sequences for usual CFTs, since invariance of a field the- allows one to arbitrarily redefine what is meant by surfaces ory under Weyl transformations for a general metric implies of constant time—with local Weyl transformations on a fixed invariance under the global conformal group when that met- class of spatial surfaces. In a word, Shape Dynamics trades ric is (conformally) flat [13]. relativity of time for relativity of scale. Each is a gauge invari- We will see that it is very useful to consider separately ance that can be defined on the phase space of general relativ- constant scalings that affect the total spatial volume, and to ity depending on one free function. The purpose of this paper distinguish these from Weyl transformations that leave the is not only to re-derive some known results through a novel spatial volume fixed. As we will see, these volume preserv- route, but also to highlight a mechanism which is an alterna- ing conformal transformations (VPCT) play a special role in tive to the AdS/CFT conjecture, which can be used to explain Shape Dynamics. some results that are usually attributed to the AdS/CFT conjecture. This mechanism is the trading of classical gauge symmetries, which we will explain in Sect. 2. The purpose 1.2 Shape Dynamics of this paper is thus to show that this mechanism is very general and that this general mechanism can be used to derive The main argument of this paper is based on number of obser- known results without using the AdS/CFT conjecture. An vations made during the development of the Shape Dynamics aspiration of the current work is that it lay down the founda- description of General Relativity, which was in part used in tions for exploring further interesting connections between [14]. These observations are: Shape Dynamics and holography. 1. There is a classical mechanism (gauge symmetry trad- 1.1 Heuristics ing), based on two partial gauge fixings of a linking gauge theory, which generates exact dualities between classical Before we posit a particular correspondence between a CFT gauge theories [15]. and a particular gravitational theory in one higher dimension, we can ask a more general question: what properties do generic gravitational theories and generic CFT’s share 2 Note that it is already known that if a classical field theory is Weyl invariant for a general metric, hab the theory will be invariant under 1 We will highlight the differences and similarities between ours and global conformal transformations when that metric is taken to be the usual holographic renormalization program when they arise. flat [13]. 123 Eur. Phys. J. C (2015) 75:3 Page 3 of 15 3

2. There is a construction principle for gauge symmetry the properties expected of a renormalized T ab of a CFT. The trading based on the generalized Stückelberg mechanism, novel feature of the derivation we will give is that the inte- also called Kretschmannization, by relativists [16]. gral of the trace of T ab must be a Weyl-invariant functional 3. The application of this mechanism to classical General of Gab, which is a necessary (but not sufficient) condition for Relativity results in a theory in which the gauge sym- it to represent a conformal anomaly. In our derivation, this is a metry due to refoliations of spacetime is traded for a direct result of the local symmetry principle of Shape Dynam- gauge symmetry of spatial Weyl transformations that ics, while in standard holographic renormalization calcula- preserve the total spatial volume of a compact Cauchy tions this result is purely coincidental unless one accepts the slice (VPCT3 symmetry). This results in Shape Dynam- broader validity of the AdS/CFT conjecture. ics, which describes gravity as a dynamical theory of Our result will be obtained in a series of steps: spatial conformal geometry [15,16]. 4. The physical predictions of the two descriptions of clas- • We explain in Sect. 3 that Euclidean AlAdS boundary sical gravity, General Relativity and Shape Dynamics, conditions imply that the r = const. slices5 are asymptot- are locally4 equivalent, i.e. the solutions to the equa- ically CMC and possess asymptotically vanishing spatial tions of motion differ only by gauge transformations. The curvature when the conformal boundary is approached. solutions coincide manifestly when General Relativity is It follows (from an argument analogous to observation evolved in constant mean (extrinsic) curvature (CMC) 6) that the radial evolution equations of General Relativ- gauge and if Shape Dynamics is evolved in a conformal ity coincide manifestly with the equations of motion of a gauge determined by the Lichnerowicz–York equation VPCT-gauge theory at the conformal boundary. This pro- [17]. vides the classical, geometric setting for the occurrence 5. General Relativity in CMC gauge is a dictionary, which of a dual CFT defined on the boundary. relates observables of General Relativity uniquely to • In Sect. 4 we derive the large volume asymptotic behav- observables of Shape Dynamics and vice versa. The local ior of the Shape Dynamics Hamilton–Jacobi functional availability of CMC gauge allows one to translate all local and recover a number of results that are obtained in physical statements from either theory into the other [18]. the holographic renormalization framework, in particu- 6. The homogeneous lapse equations of motion of Gen- lar an expression for the integral of the holographic trace eral Relativity are the CMC equations of motion in any anomaly. This is the finite piece of the integral of the CMC slice in which the spatial Ricci scalar is homo- trace of the metric’s canonical momentum, π, evaluated geneous. It follows from the manifest coincidence with at the boundary and it is the quantity that maps to the Shape Dynamics that the proper time equations of motion integral of the trace anomaly of the CFT in the AdS/CFT manifestly coincide with the equations of motion of a correspondence. VPCT-gauge theory [19]. • There is, however, an important subtlety to point out. The manifest coincidence of Shape Dynamics and Gen- A more detailed discussions of these observations follows in eral Relativity requires us to use the boundary limit of Sect. 2. the induced metric, which will be denoted by Gab below, rather than the rescaled induced metric, which we will γ (0) 1.3 Main results denote by ab below, which is used in standard presen- tations of holographic renormalization. Now, the AlAdS In this paper we consider some implications of gauge sym- boundary conditions on hab—the pullback of the space- metry trading for the class of spacetimes which provide the time metric onto foliations—imply fall-off conditions for setting for the AdS/CFT correspondence, which are called curvature invariants derived from Gab that allow one to asymptotically locally AdS spaces. Our main result is that only distinguish the integrals of these curvature invari- we will explain, without referring to the AdS/CFT conjec- ants. This has an important consequence: in even dimen- ture or any particular dual pairing of theories, why the πab sion, d, CFT’s have a trace anomaly of a local form of a spacetime diffeomorphism invariant theory of a met- [20,21] ric in the bulk has, when evaluated on the boundary of an √ = | | = , asymptotically locally AdS (Al AdS) spacetime, several of T 0 T 0 hQ (1)

where Q is a local scalar polynomial in d curvature com- 3 VPCT stands for (total) volume preserving conformal transforma- 2 tions. ponents. Since the results presented here are stated in terms of G , one cannot recover the anomaly Q(x), 4 Despite the exact local equivalence, there is a possibility for global ab differences between General Relativity and Shape Dynamics, if CMC gauge is not globally available. 5 Here the radial coordinate is deﬁned by Eq. (12), below. 123 3 Page 4 of 15 Eur. Phys. J. C (2015) 75:3

but only the integral thereof. The local counter-term the ADM description of classical gravity and explore what action and the local conformal anomaly found in stan- this exact classical equivalence implies for a partition func- dard holographic renormalization are, on the other hand, tion in the semiclassical regime. That is, we use a bottom-up γ (0) expressed in terms of ab . The restriction to Gab is approach to explain some known AdS/CFT results, rather thus the mechanism by which VPCT invariance is con- than assuming the validity of the AdS/CFT conjecture and sistent with the equivalence of General Relativity and deriving results top-down. Shape Dynamics at the boundary. In particular the AlAdS The Appendix B of this paper provides the first hints of boundary conditions imply that the integrated anomaly, how our main results could be extended to more complicated i.e. πSD, tends to a constant at the boundary and its CFTs. There, we consider the effect of a scalar field on the dependence on the boundary metric provides a non-trivial large volume behavior of the Shape Dynamics Hamilton– consistency check with the holographic renormalization Jacobi functional, where we find agreement with results from group results. Our direct calculations in Sect. 4 verify holographic renormalization. π | = SD bdy. T bdy.. An interesting extension of this Unless otherwise specified, we restrict ourselves to the work would be to redo the derivation performed here in case where both bulk and boundary metrics are Euclidean γ (0) =−d(d−1) terms of the rescaled metric ab in order to be able to and the cosmological constant, 2 is negative. compare more directly with the calculation of the local anomaly in standard approaches to holographic renor- Notation malization. We note, however, that our results are completely consistent, where applicable, with known calcu- Due to the usage of metrics of different dimensions, partic- lations. ular metric expansions, pullbacks etc., might prove slightly confusing. We thus provide the reader with a glossary of This main result can be expressed in the familiar AdS/CFT notation: language by assuming the existence of a partition function for Shape Dynamics in the bulk, which admits a recognizable6 gμν The spacetime metric on the locally asymp- semiclassical limit near the boundary. Using these assump- totically AdS manifold. tions we argue in Sect. 5 that: hab The pullback of the spacetime metric onto the hypersurfaces of the foliation. • 2 The assumption of a semiclassical limit of Shape Dynam- γab The rescaled hypersurface metrics: r γab = ics near the boundary induces equations that can be recog- hab. nized as the UV-limit of Wilsonian renormalization group Gab The asymptotic boundary value of hab (as the equations for a CFT. It is thus not necessary to assume the radius goes to infinity). AdS/CFT conjecture to explain the appearance of a CFT γ (n) γ ,γ = ab The inverse radial even expansion of ab ab at the conformal boundary; rather the manifest classical γ (n) −2n ab r equivalence of the radial evolution at the boundary with (d+1) Rabcd The full Riemann tensor restricted to spatial the Shape Dynamics evolution explains the appearance indices. 7 of a CFT near the boundary. (d) Rabcd The foliation induced hypersurface intrinsic Riemann tensor. The combination of the classical gauge symmetry trading mechanism with this assumption leads to a construction principle for dualities of AdS/CFT type. This recipe is (1) 2 Mechanism constructive and (2) very general. We thus expect that it can be applied in a wide variety of circumstances. The recipe is This section serves as an introduction to those results about summarized in Sect. 5.1. Shape Dynamics that are relevant for the construction of We also want to emphasize that our reasoning turns the exact classical dualities in the present paper. The presen- usual AdS/CFT argument upside down: we use the mathe- tation is intended to highlight: (1) the simplicity and (2) the matical bulk–bulk equivalence of the Shape Dynamics and constructive nature of our mechanism. We will be interested in applying the gauge symmetry 6 “Recognizable” means that the Ward-identities take the form of classi- trading formalism to the ADM formulation of gravity in d +1 cal constraints when momenta are substituted for functional derivatives. dimensions. This formalism is, however, very general and can 7 Note that the correspondence between bulk gravity and boundary be applied, not only to the case we are interested in here, but CFT is purely coincidental from the point of view of standard holographic renormalization arguments unless the AdS/CFT conjecture is also to other familiar models. The power of the formalism evoked. is perhaps illustrated in this paper by the fact that, in the 123 Eur. Phys. J. C (2015) 75:3 Page 5 of 15 3 context of our derivation, it constitutes the key tool which we trade the refoliation invariance generated by H[N] with explains the relation between gravity and renormalization of volume preserving Weyl transformations generated by D[ρ]. a CFT. The general formalism of gauge symmetry trading The key idea is that Shape Dynamics provides insight into was introduced in [16] and formalized in [15]. The quantum the relationship between gravity and CFT in one less dimen- aspects of these dualities were developed in [18] and some sion, because it is itself a conformal theory in one less dimen- simple examples are discussed in [22]. For convenience, we sion. This makes the existence of a correspondence between have included a technical Appendix A which summarizes the a gravity theory on M and a CFT on its boundary = ∂M general results. completely transparent because the former theory already Returning to the case of gravity, we begin with a heuristic enjoys invariance under local scale transformations on . motivation of gauge symmetry trading in General Relativity, Before getting into the technical implementation of this, then provide the technical details of the procedure. we should make one important point. Note that D[const.] is identically zero, this ensures D[ρ] only generates volume 2.1 The basic idea of gauge symmetry trading in General preserving transformations. This means that there exists an Relativity N = N0 such that H[N0] is not gauge fixed by D[ρ].This H[N0] becomes a global Hamiltonian constraint that gener- The Hamiltonian framework for General Relativity is based ates evolution on the CMC slices. on an initial phase space, , given by d dimensional metrics, πab hab,onamanifold, , and its conjugate momenta, .Phys- 2.2 Shape Dynamics description of gravity ical solutions live on a constraint surface C∈ , given by two [v]= d πab( ) kinds of first class constraints, Diff d x £vh ab, The gauge symmetry trading in gravity is formulated in terms which generate diffeomorphisms of , and Hamiltonian of d-dimensional diffeomorphism constraints Diff[v]= constraints H[N], given by the complicated expression (4) dd x πab(£vh) , which will be unaltered by the gauge + ab below. These generate refoliations of the d 1 dimensional symmetry trading mechanism, and the Hamiltonian con- spacetime. straints9 Many of the hard questions in quantum gravity and cosmology are tied to the Hamiltonian constraint. It is then of h h π cd H[N]= dd xN πab h h − ab cd √ interest that General Relativity can be reformulated in a way ac bd − | | d 1 h that trades refoliation invariance for a simpler gauge invari- kd(d − 1) +s R[h; x) − |h| . (4) ance, which is Weyl transformations on the fixed foliation. To 2 show this, we gauge fix the foliation invariance by imposing the CMC condition, Almost all of these generate on-shell refoliations of space- ab time, but, within any given foliation, at least one will generate π = habπ = const. (2) time evolution within this given foliation. To keep the remain- The gauge fixing condition can be expressed as a constraint, ing generator of time evolution, we now trade all but one of √ the H[N] for d-dimensional Weyl transformations that pre- π D[ρ]= ρ π − h , (3) serve the total d-volume. For this, we extend phase space by a V φ √ conformal factor and its canonically conjugate momentum πφ where V = h is the volume of . Now it is very impor- density , and declare this extension pure gauge by impos- π tant to note that D[ρ] generates Weyl transformations of hab ing the requirement that the φ be gauge generators. Using and πab that preserve the overall volume, V . the canonical transformation generated by When General Relativity is gauge fixed to CMC gauge, [ ] [ρ] both H N and D are imposed. We are used to thinking d ab 4 φˆ F = d x e d−2 g + φ φ , that H[N] is the generator of a gauge symmetry, while D[ρ] ab (5) is merely a gauge fixing condition. But nothing prevents these roles from being reversed. Shape Dynamics arises by inter- − 2d φ where φˆ := φ − d 2 lne d−2 and where . denotes the preting D[ρ] as the generator of a gauge symmetry which √2d φˆ is gauge fixed by H[N].8 The physics is the same—even mean taken w.r.t. |h|, and using the shorthand ˆ := e and [ρ] σ ab = πab − π ab if we choose a different gauge fixing for D . In this way d h , we find a linking theory with unsmeared

8 New results [23] show that in the phase space of ADM, there are under a number of assumptions only one set of constraints for which 9 s denotes the signature +1 for Euclidean and −1 for Lorentzian; and one can do this: the set composed of the refoliation constraint and the k, parametrize the signature and value of the cosmological constant ( − ) Weyl constraint. = k d d 1 . 22 123 3 Page 6 of 15 Eur. Phys. J. C (2015) 75:3

Hamiltonian constraints, 2.3 Special CMC slices π 2 π2 2 a b ˆ 2d H (x) = σ σ − − 1 − d−2 |h| The dictionary between the ADM description and Shape b a d − 1 d(d − 1) Dynamics is given by ADM in CMC gauge. To see this ˆ2d 2ππ 2d 2−d manifest equivalence in the equations of motion, one needs + 1 − ˆ d−2 |h| √ ( − ) | | to evolve the ADM system with the CMC lapse NCMC[h,π] dd 1 h ( − ) ( − ) which is in general a non-local functional of the canonical ˆ 4 1 d ˆ kd d 1 ˆ 2d +s R + − d−2 |h|, data h ,πab. However, if ADM data agree with R[h; x) d − 2 2 ab being a constant then N = const. is a CMC lapse and (6) hence, due to the manifest equivalence with Shape Dynam- and generators of VPCTs, ics, the ADM equations of motion with homogeneous lapse will exhibit VPCT symmetry in this slice. d 4 To show this, we first notice that the ADM Hamiltonian D[ρ]= d x ρ πφ − π −π |h| . (7) d − 2 constraints (4) imply that a CMC slice with R[h; x) = const. σ a σ b has constant b a . This implies that all coefficients of the To verify that we did not change the physical content of the |h| Lichnerowicz–York equation (9) are constants, so homo- ADM formulation, we impose the gauge fixing condition geneous is a solution, which implies that (ˆ x) ≡ 1. φ ≡ 0 which allows us to recover the ADM description of Inserting this into the defining equations (10), we see that gravity immediately. H = H(N ≡ 1). We can thus use the manifest equiva- Shape Dynamics is constructed by imposing the gauge SD lence of the ADM equations of motion with Shape Dynam- fixing condition πφ ≡ 0, which allows us to simplify the ics to predict that the homogeneous lapse evolution of the VPCT generators to ADM system in a CMC slice with homogeneous R[h; x) exhibits conformal gauge symmetry. This means, in canoni- d D[ρ]= d x ρ π −π |h| . (8) cal language, that the VPCT constraint (8) is propagated by the ADM evolution with homogeneous lapse. This can be The Hamiltonian constraints, which will be eliminated by the verified directly evolving by the VPCT constraint (8) with phase space reduction, simplify to homogeneous lapse.√ Using the fact that the√ homogeneous lapse evolution of |h| is proportional to |h| in a CMC ˆ2d 2−d ( − d) ( ) = σ aσ b √ + ˆ + 4 1 ˆ | | slice, we can verify that the VPCT constraint√ is preserved by H x b a s R h |h| d − 2 checking that ∂t π is a constant multiple of |h|: 2 π skd(d − 1) 2d + + ˆ d−2 |h|. (9) d(d − 1) 2 ∂ π = g ∂ πab + πab∂ g t ab t t ab √ ab 2 This equation is |h| ˆ times the Lichnerowicz–York equa- = 2πabπ − 3π / |h| tion [17] and thus has a unique solution for the variable ˆ . 1 ab 1 2 Since ˆ is restricted to be volume preserving, we obtain − R |h|+(πabπ − π )/ |h| , (11) 2 2 the following defining equations for the single remaining Hamiltonian constraint, which will be the generator of time √ π | | π πab/| | reparametrizations in the Shape Dynamics description of where R, h , and ab h are homogeneous by gravity: assumption. Let us conclude this section with a provocative remark. We H (x) 2d could have found conformal gauge symmetry of the equa- ˆ − √ = HSD, d 2 = 1, (10) |h|(x) tions of motion without knowing about Shape Dynamics by directly checking homogeneity of the RHS of (11). Had where HSD is independent of x. Let us conclude this sec- we found it this way, we would have wondered where the tion with the remark that the dictionary between the ADM emergent conformal symmetry came from and we might not description and Shape Dynamics is given by ADM in CMC immediately have guessed that the underlying bulk equiv- π( ) gauge, which means that √ x = const. i.e. reduced phase |h|(x) alence of the ADM formulation with Shape Dynamics is space and the equations of motion of ADM in CMC gauge responsible for the conformal symmetry. In the next section, manifestly coincide with the reduced phase space and equa- we will show that the asymptotic conformal symmetry of tions of motion of Shape Dynamics in a gauge determined by asymptotic locally AdS spaces is due to the same mechanism: ˆ φo[h,π; x) = 1, where φo[h,π; x) denotes the conformal The AlAdS boundary conditions imply that the radial CMC factor that solves the Lichnerowicz–York equation. slices have asymptotically vanishing intrinsic Ricci scalar 123 Eur. Phys. J. C (2015) 75:3 Page 7 of 15 3

10 and hence the emergent conformal symmetry at the bound- compact r = const. slices follows that limr→∞ I (r) is ary is simply the gauge symmetry of Shape Dynamics. ﬁnite in two dimensions, but diverges in three dimensions. Similar arguments can be made for relevant and marginal curvature invariants in higher dimensions, for example for (d) ab (d) −4 3 Asymptotically locally AdS and VPCT R Rab = O(r ).

The metric of a Euclidean AlAdS spacetime takes the asymptotic form (in local coordinates near the conformal boundary 4 Holographic renormalization and Shape Dynamics r →∞) 2 μ ν 2 2 a b ds = gμνdx dx = dr + r γabdx dx We will now derive the large volume approximation to the ( ) ( ) − Hamilton–Jacobi function of Shape Dynamics. These results = dr2 + r 2γ 0 dxadxb + γ 1 dxadxb + O(r 2) (12) ab ab are similar to results obtained in the Hamiltonian approach where we have denoted the rescaled spatial metric by γab, to holographic renormalization of pure gravity [7] and are a and its expansion in radii powers by γ (n). The intrinsic metric generalization of the results obtained in [14] suitable for the = . = 2γ (0) + γ (1) + O( −2) of r const slices is thus hab r ab ab r , context needed here. We conclude this section with a com- and in these types of coordinates gab = hab. parison of the two ways to derive these results and explain Moreover, it follows from the fact that AlAdS metrics why the VPCT invariance of Shape Dynamics is compatible satisfy Einstein’s equations that the Riemann tensor takes with local counter-terms and a local conformal anomaly. the asymptotic form

(d+1) 1 −2 4.1 Classical Shape Dynamics at large volume Rμνρσ =− gμρ gσν − gμσ gρν + O(r ). 2 The intrinsic components in a r = const. slice are thus In this technical section, we provide some derivations within the Shape Dynamics description of gravity. The starting point 1 (d+1) R =− (g g − g g ) + O(r −2) are the defining Eq. (10) for the Shape Dynamics Hamilto- abcd 2 ac db ad bc nian, which we will derive perturbatively in a large volume 4 =−r γ (0)γ (0) − γ (0)γ (0) + O( ) expansion. We use this Hamiltonian to find the solution to the ac db ad bc 1 (13) 2 Hamilton–Jacobi to the first two orders. The calculations for and the extrinsic curvature of r = const. slices is higher orders require a more sophisticated expansion tech- nique, which goes beyond the scope of this paper. 2 r ( ) K = γ 0 + O(1). (14) ab 2 ab 4.1.1 Volume expansion Inserting this into the Gauss–Codazzi relation for Euclidean (d+1) (d) signature Rabcd = Rabcd − KacKdb + Kad Kbc we It is convenient to isolate the d-dimensional volume and its (d) 2 find that the intrinsic Riemann tensor Rabcd = O(r ),so conjugate momentum: (d) ad bc (d) the intrinsic Ricci scalar R = h h Rabcd is = | |, = 2 π, (d) = O( −2). V h P (16) R r (15) d

= . Equation (14) implies that the r const slices are asymp- from the other degrees of freedom. For this, we define the totically CMC and Eq. (15) implies that these CMC slices fixed-volume metric and its conjugate momentum: have asymptotically homogeneous intrinsic Ricci scalar. We − 2 are thus in the special case described in Sect. 2.3 and we con- d ¯ = V , clude that the radial evolution of the ADM system becomes hab hab V0 manifestly equivalent to the Shape Dynamics evolution at the 2 V d 1 conformal boundary. The bulk VPCT symmetry of Shape π¯ ab = πab − πhab |h| , (17) Dynamics thus implies that the radial evolution exhibits V0 d VPCT symmetry at the conformal boundary. = | ¯| Let us conclude this section by looking at the behav- where V0 h is some arbitrary but fixed reference √ ( ) ior of I (r) = |h| d R[h] near the boundary in two volume. The Poisson algebra of the new variables is and three dimensions: in both cases, R scales as r −2 near the conformal boundary, but the volume element scales as 10 We work in Euclidean signature, where the boundary of d + 1AdS r 2 in two dimensions, and as r 3 in three dimensions. For is a d-sphere. 123 3 Page 8 of 15 Eur. Phys. J. C (2015) 75:3 ¯ ab • = {V, P}=1, hcd(x), π¯ (y) For n 2, we use the expansion ¯ 1 1 h(x) 2d V (c d) (d) ¯ ¯ab ˆ d−2 = + 0 ω( ) +··· , = δ aδ bδ (x − y) − hcd(y) h (x). 1 V 1 (25) 2 d V0 (18) and get This explicitly isolates the V -dependence of the theory. In 2s ˜ ˜ terms of the new variables, the defining equations (9) and H(2) =− R + (d − 1) ω(1) . (26) (10) become d Taking the mean and using integration by parts to drop d(d − 1)sk d boundary terms ( is compact without boundary) we get H =− + P2 SD 2 4(d − 1) ˜ ¯ 4(d−1) ˆ −1 ¯ ˆ 2s R s R − − H(2) =− ω(1)0 = 0. (27) + d 2 d 4 / ˆ d−2 (V/V )2 d 0 Inserting this into Eq. (26)gives σ¯ a σ¯ b a 2d + b , d−2 = , 4d 1 (19) 2 − ¯ 0 ˜ ˜ (V/V0) d 2 |h| R + (d − 1) ω(1) = 0. (28) where barred quantities and spatial averages ·0 are calcu- ¯ lated using hab. This equation admits the solution We will solve Eq. 19 by inserting the expansion ansatz ω(1) = 0. (29) ∞ ∞ −2n/d 2d −2n/d = V , ˆ d−2 = V ω , For negative Yamabe class, this solution is not unique if HSD H(n) (n) V0 V0 ˜ ˜ n=0 n=0 R happens to be in the (discrete) spectrum of . (20) • For n = 3 ... (d − 1), the same reasoning will apply. Using the result ω(n−2) = 0, we can now use the expan- and solving order by order in V −2/d . Using this expansion, sion the second line of Eq. (19) is trivially solved by 2d n ˆ d−2 = + V0 ω +··· , 1 V (n) (30) ω(0)0 = 1, ω(n)0 = 0, ∀ n ≥ 1. (21) which leads to We can now outline the procedure for finding the solution 2s ˜ ˜ order by order: H( ) =− R + (d − 1) ω − . (31) n d n 1

• For n = 0, we have trivially Taking the mean leads to: ( − ) d d 1 sk d 2 H( ) = 0, (32) H( ) =− + P . (22) n 0 2 4(d − 1) so that • For n = 1, we observe that the equations imply that the ˜ ˜ conformal factor is chosen such that R is homogeneous. R + (d − 1) ω(n−1) = 0, (33) This is known as the Yamabe problem, which has a solution on compact manifolds without boundary [24]. The which, again, has the solution equations thus ﬁx a conformal gauge (Yamabe gauge) such that ω(n−1) = 0. (34) ( ˜) = ≡ ˜. R h const. R (23) • For n = d, the solution can still easily be worked out using the previous expansions for n = d and including We indicate this section in the conformal bundle using a the σ˜ a σ˜ b term. The resulting equation is ˜ b a tilde, e.g. hab. This leads to 2s σ˜ a σ˜ b ˜ =− ˜ + ( − )˜ ω + b a . H( ) = s R,ω( ) = 1. (24) H(d) R d 1 (d−1) (35) 1 0 d |h˜| 123 Eur. Phys. J. C (2015) 75:3 Page 9 of 15 3 Taking the mean, we get d(d − 1)sk d δS 2 0 =− + 2 4(d − 1) δV σ˜ a σ˜ b b a 2/d δ δ 2 H(d) = . (36) ˜ V0 S S V0 | ˜| +s R + +··· (42) h ˜ ˜ab V δhab δh V order by order in V using the ansatz ω(d−1) can then be solved by inverting the following equation: ∞ 1− 2n V d S = S( ), (43) V0 n σ˜ a σ˜ b n=0 d b a cω( − ) − d 1 2s(d − 1) | ˜| h for odd d or σ˜ a σ˜ b d b a ∞ = cω(d−1) − , (37) 1− 2n 2s(d − 1) | ˜| V V d h S = log S( d ) + S(n), (44) V0 2 V0 = d n 2 where c is the d-dimensional conformal Laplacian, for even d (because the previous ansatz is not valid with even ˜ d). ˜ R c = + . (38) (d − 1) We can get a recursion relation for the solution by taking the asymptotic boundary condition Thus, all higher order terms will be non-local because lim S = S(0) = const. (45) they will involve inverting the conformal Laplacian. (V/V0)→∞

Collecting the ﬁrst three non-zero terms, we get Using this we get: / ( − ) 2 d • = =− d d 1 sk + d 2 + ˜ V0 For n 0, the solution is trivial HSD 2 ( − ) P s R l 4 d 1 V √ ( − ) σ˜ a σ˜ b 2 2 −4 d 1 a d S( ) =± −sk . + b V0 + O V . (39) 0 2 (46) |h¯| V V0 • For n = 1, the solution is equally straightforward. The 4.1.2 Hamilton–Jacobi equation result for d = 2is

˜ Using the substitutions s R S(1) =± √ . (47) (d − 2) −sk δS P → (40) In d = 2 this term gives the conformal anomaly. It is δV found to be and the fact that, for a VPCT-invariant S, one can use11 ˜ = s R Sd 2 =± √ . (48) (1) − δ δ 2 sk σ˜ ab → S − 1 ˜ S ãb | ˜|, ˜ hab ˜ h h (41) δhab d δhab • For n = 2, we use ab δ δ 2 ˜2 where S = S(hab,α ) is the HJ functional, depending on S(1) S(1) s ãb ˜ R =− R Rab − , ( + )/ αab ˜ ãb ( − )2 the metric hab and on d d 1 2 integration constants , δhab δh k d 2 d we can solve the Hamilton–Jacobi equation associated to (49) Eq. (39), and find 11 Notice that the second substitution is by no means trivial: it involves √ 3 ab 2/d δS P ab δS s the calculation π¯ = (V/V0) − hh = ¯ ,using δhab 2 δh S( ) =∓√ ab√ 2 2 δS −2/d δS δS δV δV 1 ab −sk (d − 4)(d − 2) the fact that δ = (V/V0) ¯ + δ δ and δ = hh . hab δhab V hab hab 2 Furthermore, it requires the realization that, for a VPCT-invariant S, ãb ˜ d ˜2 × R Rab − R . (50) one can ignore the variation of the Yamabe conformal factor. 4(d − 1) 123 3 Page 10 of 15 Eur. Phys. J. C (2015) 75:3

In d = 4 this gives the anomaly. It is 2. In the large volume limit, the divergent terms, which are integrals of local terms for Yamabe metrics, are turned 3 into non-local terms for metrics that are not Yamabe. d=4 s ˜ab ˜ 1 ˜2 S( ) =∓√ R Rab − R . (51) 3. The integrated form of the anomaly is manifestly VPCT 2 −sk 8 3 in its local form and hence has the same form in any conformal gauge. • For n > 2, If we ignore higher order terms in the V -

expansion of HSD then we get a compact recursion relation We have to address the subtle issue we mentioned in for S(n), the introduction regarding the difference between the local anomaly, Eq. (1) that appears in a CFT and the integrated form p+q=n δS(p) δS(q) of the anomaly that arises in Shape Dynamics. The bulk– S(n) =± √ (d − n) −sk δ ˜ δ ãb bulk equivalence between the ADM and the Shape Dynam- 2 p,q>0 hab h ics description of General Relativity raises the question of (d − 2p)(d − 2q) − S( ) S( ) + .... how this discrepancy between the ADM and Shape Dynam- ( − ) p q (52) 4 d 1 d ics descriptions is possible? The answer to this question lies in the following: despite In general, though, there will be contributions from the fact that all observables of Shape Dynamics coincide with higher order terms that depend on the inverse of but observables of the ADM description, the Shape Dynamics these are not important for d < 5. equations of motion coincide with the ADM equations of motion only in the dictionary, i.e. only when the ADM sys- 4.2 Comparison with holographic renormalization results tem is in a CMC slice and evolved with the CMC lapse. We saw in Sect. 3 that the radial evolution of the ADM system The Hamiltonian approach to holographic renormalization satisfied these conditions only at the conformal boundary, [7] uses the AdS/CFT correspondence to relate the near but that these conditions are violated at any distance from boundary behavior of the classical Hamilton–Jacobi func- the boundary. This provides us with a non-trivial consistency tional of a gravity theory to the partition function of a CFT check: the integrals of the non-local VPCT-invariant terms in the strong coupling limit. In particular, the radial evolu- derived in the large volume expansion of Shape Dynamics tion near the conformal boundary of the Euclidean AlAdS have to coincide with the integrals of the local terms derived Hamilton–Jacobi functional, S, is related to the renormaliza- from the radial ADM evolution at the boundary. tion of a dual CFT on the boundary: in the large volume limit The explicit evaluation of the curvature invariants at the V →∞, the divergent part of S is identified with counter- boundary (performed at the end of Sect. 3) confirm this asser- terms, while a standard argument shows that the logarith- tion. For example, in d = 2, the VPCT-invariant Shape mic term can be identified with the integral of the confor- Dynamics anomaly falls off as 12 mal anomaly. These terms are local in the sense that the 1 (d) |h| R[h]=O(r −2) counter-term Lagrangian contains local curvature invariants. V The results of the previous subsection show that the large volume limit of the Shape Dynamics Hamilton–Jacobi func- which is the same fall-off rate for the local anomaly when tional takes the same form as the results from General Rela- expressed in terms of the induced boundary metric Gab γ (0) tivity described in [7] if the conformal factor of the intrinsic (rather than the rescaled boundary metric ab used in standard holographic renormalization). This ensures that the inte- metric hab is replaced with the Yamabe conformal factor. This choice of conformal frame has three important con- grals coincide at the boundary. The same mechanism works sequences for the Shape Dynamics Hamilton–Jacobi func- for the higher dimensional counter-terms; again using Gab, tional: we find for example ãb ˜ 1 ˜2 −4 ab 1 2 R Rab − R = O(r ) = R Rab − R . 1. It ensures that the Shape Dynamics Hamilton–Jacobi 3 3 functional is manifestly VPCT invariant, i.e. invariant This shows how the VPCT invariance of Shape Dynamics under d-dimensional Weyl transformations that preserve is compatible with local counter-terms in the radial ADM the total d-volume. evolution: The equations of motion for Gab of General Rel- ativity and Shape Dynamics coincide manifestly when the boundary is approached. Since the boundary metric G 12 A slightly different procedure allows for the calculation of the local ab form of the anomaly [3], which is consistent, in the near boundary limit, gives only access to the integrals of the curvature invariants, to the results presented here for the reasons described below. one might think that Shape Dynamics provides less infor- 123 Eur. Phys. J. C (2015) 75:3 Page 11 of 15 3 mation than standard holographic renormalization, which is tivity and Shape Dynamics and explore the consequence of γ (0) expressed in terms of the rescaled metric ab , which provides this equivalence for an assumed gravity partition function access to local curvature invariants. However, Shape Dynam- with a recognizable semiclassical limit near the conformal ics provides actually more, once one moves away form the boundary. boundary. It states that if we replaced radial evolution into We start with assuming the existence of a Shape Dynamics ¯ ¯ the bulk with CMC evolution into the bulk, then we would boundary amplitude ZV [h], where hab denotes the rescaled not only find VPCT invariance at the boundary, but also in metric at CMC volume V . This boundary amplitude is sup- the bulk. posed to be obtained as the solution to the semiclassical We can make two further remarks on the results found so defining equations of Shape Dynamics in a Euclidean AlAdS far. space in the limit V →∞. We suppose that the semiclassical limit is recognizable in the sense that the classical • It should be emphasized that in the case that the bulk Shape Dynamics constraints are turned into operators act- δ ing on Z [h¯] through the replacement πab(x) → ih¯ . manifold is pure Anti-de Sitter spacetime (rather than V δhab(x) AlAdS), the metric and curvatures are homogeneous and Since the boundary conditions imply that the classical Shape the constant r slices are CMC in the bulk. In this case, the Dynamics Hamiltonian constraint asymptotes into the homo- Hamilton–Jacobi functions of Shape Dynamics coincides geneous lapse Hamiltonian in the limit V →∞, we impose with that of General Relativity exactly in the bulk as well in the large V limit the radial Wheeler–DeWitt equation: as the boundary. 1 δ2 • Our results indicate the emergence of full Weyl invariance dd x −h¯ 2 h h − h h |h| +··· ac bd − ab cd δ δ on the boundary as the volume dependence disappears d 1 hab hcd ¯ when the limit V →∞is taken. That is, after removal ×ZV [h]=0 + O(h¯ ). (53) of the divergent terms, the remaining finite terms in S are fully Weyl invariant in the limit. However, it is well Moreover, the classical VPCT constraints of Shape Dynam- known that a classical field theory defined on a manifold ics lead to with an arbitrary metric, hab, that is Weyl invariant will δ have global conformal symmetry under SO(2, d) when h (x) Z [h¯]=0 + O(h¯ ). (54) ab δ ¯ ( ) V that metric is taken to be flat [13].13 This will apply to hab x the finite parts of S in the V →∞limit. That func- Equation (54) has an important consequence: it states that tion then has several properties needed to describe the the wrong sign in the kinetic term of Eq. (53)isO(h¯ ), except semiclassical limit of a CFT. for the derivative w.r.t. V . We can thus rewrite Eq. (53)asa second order evolution equation in V with a positive definite √ 2 ¯ d 2 δ ZV [h] kinetic term d xh¯ hachbd/ |h| . This term looks 5 Remarks on Wilsonian renormalization δhabδhcd like the Schwinger–Dyson equation for a mass term in d So far, we used the classical bulk equivalence of General Rel- dimensions, which is the UV-limit of an IR suppression term ativity and Shape Dynamics to explain why classical gravity as it is used in the exact renormalization group framework; ... is related to a classical conformal field theory near the confor- see e.g. [25]. If we interpret the extra terms in Eq. (53) mal boundary of AlAdS spaces. This is, however, not how the as the remnant of a particular scheme, then we have an argument that relates the radial evolution with renormalization AdS/CFT correspondence is used in practice, where the rela- 14 tion between the radial evolution of classical gravity and the group flow. We want to warn the reader that this heuris- RG flow of a CFT in the strong coupling limit is used. We will tic argument is at best a plausibility argument: the validity now present a heuristic observation that can be used to turn of our assumptions have to be checked before the argument the classical correspondence we have so far described into a worked out in a particular application. However, we want to quantum correspondence. The purpose of this section is thus point out that the VPCT invariance of Shape Dynamics is to turn the AdS/CFT logic upside down, as was suggested the essential ingredient that allows us to reinterpret radial + in [19]. Whereas the usual logic assumes the AdS/CFT con- evolution of semiclassical gravity in AlAdS d 1asthe jecture and derives the holographic renormalization group equations in a semiclassical limit, we go the other way: we 14 Notice the important difference that usual renormalization group use the proven classical equivalence between General Rela- equations are first order equations, while we derive a second order equation in V . In a semiclassical regime, one can argue that ZV willbeasum of an “incoming” and an “outgoing” wave function in V . The interpre- 13 This can be extended to a conformally flat metric (as opposed to flat), tation of our second order equation as a renormalization group equation by using the complete set of conformal Killing vector fields in place of requires that we restrict ourselves to the “outgoing” summand. This was the coordinates in flat space. also discussed by Freidel [9]. 123 3 Page 12 of 15 Eur. Phys. J. C (2015) 75:3

UV-limit of renormalization group flow of a classical CFT in cients found for the integrals of trace anomalies using the d dimensions. methods of holographic renormalization [3,7]. If we interpret the correspondence between radial evolu- Some concluding remarks are in order: tion of Shape Dynamics and Wilsonian renormalization liter- ally, then we might be able to reconcile the non-local VPCT- • We provide a mathematical mechanism (trading of gauge invariant counter-terms that found in the present paper with symmetries) as a construction principle for classical dual- the usual local counter-terms. This is due to the fact that the ities. This mechanism provides a complete one-to-one fixed point of a Wilsonian flow equation may differ from the dictionary between the physical predictions of two at first critical bare action by a scheme dependent one-loop deter- sight very different looking gauge theories. Moreover, the minant (see e.g. [26]). The reconciliation would thus follow dictionary proves that all local physical predictions of from the conjecture that the radial evolution of Shape Dynam- these two classical gauge theories coincide. The symme- ics is equivalent to a particular Wilsonian scheme. try trading mechanism has previously been used to show that spacetime General Relativity is physically equiva- 5.1 Generic recipe lent to Shape Dynamics by trading refoliation invariance of spacetime for local spatial conformal invariance. The combination of the heuristic quantum argument pre- • In this paper we showed that the classical bulk–bulk sented in this section with the classical symmetry trading equivalence of Shape Dynamics and General Relativity provides a generic mechanism that can be used to construct explains some aspects of classical AdS/CFT, in particu- dualities of the type of holographic renormalization. This lar conformal symmetry of the boundary theory and the recipe can be summarized as follows: form of the classical Hamilton–Jacobi functional. How- ever, Shape Dynamics does not explain specific corre- 1. Use the classical linking theory formalism to derive a spondences between particular CFT’s and their dual grav- bulk–bulk equivalence between two classical gauge the- itational theories. We note that our results reproduce, but ories. do not predict, the correspondence between pure Gen- 2. Construct the dictionary between these two classical eral Relativity and N = 4 super-Yang–Mills theory in gauge theories. the N →∞limit which was found by holographic renor- 3. Assume a partition function with a recognizable semi- malization group methods [3,7]. classical limit in dictionary gauge. Then use the classical • A possible extension of the work in this paper is to dictionary to reinterpret the semiclassical quantum equa- explore the conjecture that the correspondence we have tions. demonstrated here extends to a stronger correspondence between a quantization of Shape Dynamics and a quan- We believe in the value of this generic recipe, in part because tum CFT. The evidence we possess for an extension of the the calculations in Sect. 4 show that this generic recipe can correspondence into the quantum regime is the apparent be used to find a number of results that are usually attributed absence of anomalies (for odd dimensions) of our spa- to the AdS/CFT correspondence. tial conformal transformations and the expected effect of matter fields (see Appendix B). In spite of their classical correspondence, the quantization of Shape Dynamics 6 Conclusions is unlikely to coincide with a quantization of General Relativity due to the very different structure of their con- In this paper we have shown that the Hamilton–Jacobi func- straints. tion of Shape Dynamics has, when evaluated on the boundary • The gauge symmetry trading mechanism is very generic. of an Al AdS spacetime, with divergent terms removed, sev- We thus expect that it can be successfully applied to eral properties needed to posit a correspondence to the semi- the attack of a variety of problems that are not part of classical limit of the effective action of a CFT. By invoking AdS/CFT. A first example of this sort was explored in gauge symmetry trading this explains very generally why the [22], where the U(1)-gauge symmetry of classical elec- foliation invariance of General Relativity manifests itself as tromagnetism was traded for shift symmetry. A main conformal invariance on the boundary of an Al AdS space- motivation for this paper was to advertise the power of the time. Thus, gauge symmetry trading provides a deep and gen- symmetry trading mechanism to researchers interested in eral explanation of why there exist correspondences between dualities. gravitational theories invariant under spacetime diffeomorphisms and conformal field theories in one lower dimension. Acknowledgments We are grateful to Julian Barbour for encourage- As a check on the general argument we also confirmed that ment, to Kostas Skenderis for comments on a draft of this manuscript Shape Dynamics reproduces the precise forms and coeffi- and to Paul McFadden for helpful conversations. We also thank Jacques 123 Eur. Phys. J. C (2015) 75:3 Page 13 of 15 3

Distler and Laurent Freidel for pointing out an oversight in the first of the two theories differ. In other words: the gauge gener- draft of this paper. S.G.’s work was supported by the Natural Science ators σα(p, q) of (1) can be traded for the gauge generators and Engineering Research Council (NSERC) of Canada and by the ρα(p, q) without changing any physical prediction of the Netherlands Organization for Scientific Research (NWO) (Project No. 620.01.784). Research at Perimeter Institute is supported by the Gov- theory. ernment of Research at Perimeter Institute is supported by the Govern- ment of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This A.2. Dictionary and observable equivalence research was also partly supported by grants from NSERC, FQXi and the John Templeton Foundation. The manifest equivalence of the two descriptions can be seen by further gauge fixing: imposing ρα(p, q) ≡ 0asa Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and gauge fixing condition on (1) and working out the phase reproduction in any medium, provided the original author(s) and the space reduction gives precisely the same reduced phase source are credited. space red and Dirac bracket {., .}D as it is obtained by 3 Funded by SCOAP /LicenseVersionCCBY4.0. imposing σα(p, q) ≡ 0 on (2). This reduced gauge theory 3 ( red, {., .}D, Hred, {χν }ν∈N ) serves as a dictionary between the two gauge theories, where the two descriptions manifestly Appendix A: general mechanism for gauge symmetry coincide. trading A constructive procedure to construct the dictionary goes as follows: consider the Poisson algebra of observables of the A.1. Symmetry trading and linking gauge theories linking theory. We can then use the fact that the phase space reduction is a Poisson-isomorphism for observables15 to cal- A (special) linking gauge theory is a canonical gauge theory culate entries of the dictionary one-by-one: pick an observ- ( , {., .}, H, {χμ}μ∈M), where the phase space and Pois- able O of the linking theory and insert the two phase space son structure ( , {., .}) can be written as a direct product reductions. Thus every linking theory observable O relates of two mutually commuting phase spaces ( o, {., .}o) and an observable of (1) with an observable of (2) by ( e, {., .}) and where the first class (coisotropic) constraint C ={ ∈ : χ ( ) = ,μ ∈ M} surface x μ x 0 can be specified (1) (2) O := O|(φ≡ ,π≡π ( , )) ↔ O O|(φ≡φ ( , ),π≡ ) , by three disjoint sets of irreducible constraints, 0 o p q o p q 0 (56) 1 χα = φα − σα(p, q), χ α = ρα( , ) − π α, (i) 2 p q (55) where the O denote an observable of the system (i). Notice 3 3 that this procedure is surjective and a Poisson-isomorphism χν = χν (p, q) + more, for observables, i.e. if one starts with a complete observable α where more vanishes when either φα ≡ 0orπ ≡ 0 holds. algebra of the linking theory, one obtains a complete dictio- (p, q) denote here local Darboux coordinates for ( o, {., .}o), nary, which is given by a Poisson-isomorphism between the while φα denote local position coordinates on ( e, {., .}e), observable algebras of description (1) and description (2). whose canonically conjugate momenta are π α. A canonical gauge theory of this kind can be gauge fixed in two A.3. Construction of linking gauge theories very interesting ways: (1) by imposing φα ≡ 0 and a par- χ tial gauge fixing condition for the constraints 2 and (2) by A useful construction principle for linking gauge theories π α ≡ imposing 0 as a partial gauge fixing condition for is Kretschmannization (also called the general Stückelberg χ the constraints 1. The two partial gauge fixings lead to the mechanism), which is most simply explained for the action same reduced phase space o and the Dirac bracket associ- of an Abelian group G on configuration space. Let us denote ated with this phase space reduction reduces to {., .}o, i.e. local coordinates on configuration space by qi and consider the Poisson bracket on o. The two theories have, however, an Abelian group action g(φ): qi → Qi (q,φ)that is locally distinct constraints and Hamiltonians. parametrized by group parameters φα. We can implement The partial gauge fixing (1) reduces the constraints to gauge invariance under this group action in any gauge theory 3 σα(p, q) and χμ(p, q) and the Hamiltonian is H(p, q,σα ( o, {., .}o, Ho, {χμ}μ∈M) using the following steps: (p, q), 0) ≈ H(p, q, 0, 0). The partial gauge fixing (2) on the other hand reduces the constraints to ρα(p, q) and 3 α 15 χμ(p, q) and the Hamiltonian is H(p, q, 0,ρ (p, q)) ≈ An observable is an equivalence class of gauge-invariant phase space H(p, q, 0, 0). The two gauge theories describe the same functions, where two phase space functions are called equivalent if they coincide on the constraint surface. For simplicity, we will slightly abuse physics, since each of them is obtained as a partial gauge notation and write representative phase space functions O rather than fixing of the same linking theory, but the gauge symmetries equivalence classes [O]∼. 123 3 Page 14 of 15 Eur. Phys. J. C (2015) 75:3

1. Extend phase space with the cotangent bundle of the Adding this to the Hamiltonian constraint and perform- group, which we will denote by ( e, {., .}), and declare ing the phase space extension and canonical transformation the extension to be pure gauge by introducing the auxil- described in the main text, we obtain the Shape Dynamics iary first class constraints π α ≈ 0, where the π α denote Hamiltonian coupled to a scalar field, the momenta canonically conjugate to the group param- ( − ) d d 1 sk d 2 eters φα. H =− + P + U(ϕ) SD l2 4(d − 1) 2. Perform the “Kretschmannization” canonical transfor- i α − 4 4(d − 1) F = Q (q,φ)P + φα ˆ − ˜ ¯ab ¯ ¯ ˆ −1 2 ˆ mation generated by i , where +s d 2 R − h ∇aϕ∇bϕ − ∇ ¯ d − 2 g capital letters denote the transformed variables. This 2/d 2 canonical transformation changes the form of the Hamil- V0 − 4d 1 V0 × + ˆ d−2 σ¯ a σ¯ b + π 2 . ( , ) → ( , ,φ) a ϕ (59) tonian Ho p q H p q and the gauge genera- V g¯ b V tors χμ(p, q) →˜χμ(p, q,φ), but most importantly, it In [27], it was shown that a consistent coupling of mat- implies that the auxiliary gauge generators transform as ter to Shape Dynamics requires that the matter fields be invariant under VPCT. They can, however, transform non- π α → π α − i α( ), p Wi q (57) trivially under homogeneous conformal transformation, and the weight of this transformation represents the anomalous α( )∂i scaling, , of the matter fields. For a real scalar field, we where the Wi q are the vector fields that generate the G action on configuration space. If now a subset find that the dilatation operator has the following action in {˜χα(p, q,φ)}α∈A of the transformed gauge generators the extended theory: χ˜μ(p, q,φ)can be uniquely solved for the φα, then we have a conformal linking gauge theory. To see which theories are linked let us, δρ ϕ(x) = ( − d) ρ ϕ(x). (60) without loss of generality, assume that the group parametriza- tion is such that φα ≡ 0 denotes the unit element of G, The volume dependence can then be extracted using the so Qi (q, 0) = qi . Then imposing the gauge fixing con- canonical transformation (ϕ, πϕ) → (ϕ,¯ π¯ϕ): φ ≡ dition α 0 and working out the phase space reduction reduces the theory to the system ( , {., .} , H , {χμ}μ∈M) 2 −1 −2 −1 o o o V d V d we started with. On the other hand, imposing the gauge fixing ϕ = ϕ,¯ πϕ = π¯ϕ. (61) V V condition π α ≡ 0 yields a system in which the gauge gen- 0 0 χ˜ ( , ) erators α p q have been traded for the gauge generators The volume expansion of the Shape Dynamics Hamilto- i α( ) G p Wi q , which implement the action we have put into nian can now be performed. In general, consistency of the our construction. equations will limit the possibilities for U(ϕ) for a particular We thus have a very generic construction principle: we can anomalous scaling, . Because the volume expansion will, start with an arbitrary gauge theory and the mechanism allows in general, depend on ϕ and πϕ, the volume expansion of χ ( , ) us to trade a subset of its gauge generators α p q for a the Hamilton–Jacobi equation will be modified. This will, in different set of gauge symmetry generators without changing turn, affect the expansion coefficients of the volume expan- the physical description. The only non-trivial requirement is sion. χ˜ ( , ,φ) that the Kretschmannized gauge generators α p q can For a simple illustration of this, consider the case where φ be uniquely solved for the α. = d. In this case, ϕ has no conformal scaling. The zeroth- order equation is ( − ) Appendix B: scalar field (d ≤ 4) d d 1 sk d 2 H =− + P + U(ϕ( )) , (62) (0) l2 4(d − 1) 0 The goal of this appendix is to illustrate how, in the presence of matter fields, the coefficients in the volume expansion of which is only consistent and non-trivial (i.e., ϕ = const) if (ϕ) = Hamilton’s principal function, S[hab], for Shape Dynamics is the scalar field is free so that U 0. We simplify the will depend upon the matter fields. We will demonstrate this calculation with the gauge choice using the simple example of a single real scalar field, ϕ. ¯ ¯ ¯ ¯ ˜ The matter Hamiltonian for this scalar field is R − hab∇aϕ∇bϕ = const ≡ Rϕ. (63) π 2 ϕ ab Then the first order equations lead to Hm(ϕ, hab) = √ + U(ϕ) − sh ∇aϕ∇bϕ |h|. |h| ˜ (58) H(1) = s Rϕ,ω(1) = 1. (64) 123 Eur. Phys. J. C (2015) 75:3 Page 15 of 15 3

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